Lost in Aggregation: The Causal Interpretation of the IV Estimand

TL;DR

The paper shows that when the treatment is an aggregate of multiple components with heterogeneous effects, the aggregate causal effect ACE(A,Y) is not well-defined without specifying how interventions on the aggregate are realized through its components (ACID). It formalizes ACID, defines ACE under a valid ACID, and derives the IV estimand in a linear Gaussian SCM as $\beta_{IV}=\frac{\sum_j \beta_j\delta_j}{\sum_j \alpha_j\delta_j}$. It identifies two regimes—proportional aggregation and instrument-tuned interventions—that guarantee IV identifies ACE; otherwise, IV may diverge from ACE, as demonstrated by simulations. The work also discusses exclusion, Sargan tests, and extensions to aggregated outcomes or instruments, highlighting substantial cautions for interpreting IV results with aggregated treatments. Overall, it urges explicit aggregation-justification and careful consideration of ACID assumptions when using IV for policy guidance in settings with aggregate treatments.

Abstract

Instrumental variable based estimation of a causal effect has emerged as a standard approach to mitigate confounding bias in the social sciences and epidemiology, where conducting randomized experiments can be too costly or impossible. However, justifying the validity of the instrument often poses a significant challenge. In this work, we highlight a problem generally neglected in arguments for instrumental variable validity: the presence of an ''aggregate treatment variable'', where the treatment (e.g., education, GDP, caloric intake) is composed of finer-grained components that each may have a different effect on the outcome. We show that the causal effect of an aggregate treatment is generally ambiguous, as it depends on how interventions on the aggregate are instantiated at the component level, formalized through the aggregate-constrained component intervention distribution. We then characterize conditions on the interventional distribution and the aggregate setting under which standard instrumental variable estimators identify the aggregate effect. The contrived nature of these conditions implies major limitations on the interpretation of instrumental variable estimates based on aggregate treatments and highlights the need for a broader justificatory base for the exclusion restriction in such settings.

Figures (7)

• Figure 1: The aggregate setting: Comparison of the classic IV setting as posited by the researcher (top) and the true data generating process (bottom).
• Figure 2: A: IV estimates for different samples sizes $n$ and edge weights $\beta_1$. The vertical purple dashed line indicates proportional aggregation. B: IV estimates for different sample sizes $n$, and ACID slopes $d_1$. The vertical orange dash-dotted line denotes the instrument-tuned intervention. In both plots, $\beta_\text{IV}$ is denoted by the yellow line and the $\text{ACE}(A,Y)$ is denoted by the grey line. The IV estimates are denoted by: red circles ($n=10$), green triangles ($n =100$), blue diamonds ($n=1000$).
• Figure 3: A violation of the exclusion restriction that is equivalent to the aggregate setting under linearity and Gaussianity.
• Figure 4: Empirical power of the Sargan test for two strong instruments (red), one strong and one weak instrument (green), and two weak instruments (blue) as a function of deviations from proportional aggregation for A: type-I-error $0.01$, B: type-I-error $0.5$. $H_0$ is true for $\beta_1=2$ (marked with the purple dashed line).
• Figure 5: Distribution of $F$-statistics for pairs of instruments of different strengths. The typical cut-off of 11 is marked in yellow where applicable.

Theorems & Definitions (2)

• Definition 1: Aggregate Causal Effect
• Proposition 1: Aggregation as an exclusion violation